##### 4.15.3 $$y(x) \left (1-2 x^3 y(x)\right )+x \left (1-2 x y(x)^3\right ) y'(x)=0$$

ODE
$y(x) \left (1-2 x^3 y(x)\right )+x \left (1-2 x y(x)^3\right ) y'(x)=0$ ODE Classiﬁcation

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.325852 (sec), leaf count = 358

$\left \{\left \{y(x)\to \frac {\sqrt [3]{2} \left (-x^3+c_1 x\right )}{\sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}+\frac {\sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}{3 \sqrt [3]{2} x}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^3-c_1 x\right )}{2^{2/3} \sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}{6 \sqrt [3]{2} x}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^3-c_1 x\right )}{2^{2/3} \sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-27 x^2+\sqrt {729 x^4+108 x^3 \left (x^3-c_1 x\right ){}^3}}}{6 \sqrt [3]{2} x}\right \}\right \}$

Maple
cpu = 0.137 (sec), leaf count = 522

$\left [y \left (x \right ) = \frac {\left (\left (-108+12 \sqrt {12 x^{8}-36 \textit {\_C1} \,x^{6}+36 x^{4} \textit {\_C1}^{2}-12 \textit {\_C1}^{3} x^{2}+81}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {6 \left (\frac {x^{2}}{3}-\frac {\textit {\_C1}}{3}\right ) x}{\left (\left (-108+12 \sqrt {12 x^{8}-36 \textit {\_C1} \,x^{6}+36 x^{4} \textit {\_C1}^{2}-12 \textit {\_C1}^{3} x^{2}+81}\right ) x^{2}\right )^{\frac {1}{3}}}, y \left (x \right ) = -\frac {\left (\left (-108+12 \sqrt {12 x^{8}-36 \textit {\_C1} \,x^{6}+36 x^{4} \textit {\_C1}^{2}-12 \textit {\_C1}^{3} x^{2}+81}\right ) x^{2}\right )^{\frac {1}{3}}}{12 x}+\frac {3 \left (\frac {x^{2}}{3}-\frac {\textit {\_C1}}{3}\right ) x}{\left (\left (-108+12 \sqrt {12 x^{8}-36 \textit {\_C1} \,x^{6}+36 x^{4} \textit {\_C1}^{2}-12 \textit {\_C1}^{3} x^{2}+81}\right ) x^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (-108+12 \sqrt {12 x^{8}-36 \textit {\_C1} \,x^{6}+36 x^{4} \textit {\_C1}^{2}-12 \textit {\_C1}^{3} x^{2}+81}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}+\frac {6 \left (\frac {x^{2}}{3}-\frac {\textit {\_C1}}{3}\right ) x}{\left (\left (-108+12 \sqrt {12 x^{8}-36 \textit {\_C1} \,x^{6}+36 x^{4} \textit {\_C1}^{2}-12 \textit {\_C1}^{3} x^{2}+81}\right ) x^{2}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (\left (-108+12 \sqrt {12 x^{8}-36 \textit {\_C1} \,x^{6}+36 x^{4} \textit {\_C1}^{2}-12 \textit {\_C1}^{3} x^{2}+81}\right ) x^{2}\right )^{\frac {1}{3}}}{12 x}+\frac {3 \left (\frac {x^{2}}{3}-\frac {\textit {\_C1}}{3}\right ) x}{\left (\left (-108+12 \sqrt {12 x^{8}-36 \textit {\_C1} \,x^{6}+36 x^{4} \textit {\_C1}^{2}-12 \textit {\_C1}^{3} x^{2}+81}\right ) x^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (-108+12 \sqrt {12 x^{8}-36 \textit {\_C1} \,x^{6}+36 x^{4} \textit {\_C1}^{2}-12 \textit {\_C1}^{3} x^{2}+81}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}+\frac {6 \left (\frac {x^{2}}{3}-\frac {\textit {\_C1}}{3}\right ) x}{\left (\left (-108+12 \sqrt {12 x^{8}-36 \textit {\_C1} \,x^{6}+36 x^{4} \textit {\_C1}^{2}-12 \textit {\_C1}^{3} x^{2}+81}\right ) x^{2}\right )^{\frac {1}{3}}}\right )}{2}\right ]$ Mathematica raw input

DSolve[y[x]*(1 - 2*x^3*y[x]) + x*(1 - 2*x*y[x]^3)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (2^(1/3)*(-x^3 + x*C[1]))/(-27*x^2 + Sqrt[729*x^4 + 108*x^3*(x^3 - x*C
[1])^3])^(1/3) + (-27*x^2 + Sqrt[729*x^4 + 108*x^3*(x^3 - x*C[1])^3])^(1/3)/(3*2
^(1/3)*x)}, {y[x] -> ((1 + I*Sqrt[3])*(x^3 - x*C[1]))/(2^(2/3)*(-27*x^2 + Sqrt[7
29*x^4 + 108*x^3*(x^3 - x*C[1])^3])^(1/3)) - ((1 - I*Sqrt[3])*(-27*x^2 + Sqrt[72
9*x^4 + 108*x^3*(x^3 - x*C[1])^3])^(1/3))/(6*2^(1/3)*x)}, {y[x] -> ((1 - I*Sqrt[
3])*(x^3 - x*C[1]))/(2^(2/3)*(-27*x^2 + Sqrt[729*x^4 + 108*x^3*(x^3 - x*C[1])^3]
)^(1/3)) - ((1 + I*Sqrt[3])*(-27*x^2 + Sqrt[729*x^4 + 108*x^3*(x^3 - x*C[1])^3])
^(1/3))/(6*2^(1/3)*x)}}

Maple raw input

dsolve(x*(1-2*x*y(x)^3)*diff(y(x),x)+(1-2*x^3*y(x))*y(x) = 0, y(x))

Maple raw output

[y(x) = 1/6/x*((-108+12*(12*x^8-36*_C1*x^6+36*_C1^2*x^4-12*_C1^3*x^2+81)^(1/2))*
x^2)^(1/3)-6*(1/3*x^2-1/3*_C1)*x/((-108+12*(12*x^8-36*_C1*x^6+36*_C1^2*x^4-12*_C
1^3*x^2+81)^(1/2))*x^2)^(1/3), y(x) = -1/12/x*((-108+12*(12*x^8-36*_C1*x^6+36*_C
1^2*x^4-12*_C1^3*x^2+81)^(1/2))*x^2)^(1/3)+3*(1/3*x^2-1/3*_C1)*x/((-108+12*(12*x
^8-36*_C1*x^6+36*_C1^2*x^4-12*_C1^3*x^2+81)^(1/2))*x^2)^(1/3)-1/2*I*3^(1/2)*(1/6
/x*((-108+12*(12*x^8-36*_C1*x^6+36*_C1^2*x^4-12*_C1^3*x^2+81)^(1/2))*x^2)^(1/3)+
6*(1/3*x^2-1/3*_C1)*x/((-108+12*(12*x^8-36*_C1*x^6+36*_C1^2*x^4-12*_C1^3*x^2+81)
^(1/2))*x^2)^(1/3)), y(x) = -1/12/x*((-108+12*(12*x^8-36*_C1*x^6+36*_C1^2*x^4-12
*_C1^3*x^2+81)^(1/2))*x^2)^(1/3)+3*(1/3*x^2-1/3*_C1)*x/((-108+12*(12*x^8-36*_C1*
x^6+36*_C1^2*x^4-12*_C1^3*x^2+81)^(1/2))*x^2)^(1/3)+1/2*I*3^(1/2)*(1/6/x*((-108+
12*(12*x^8-36*_C1*x^6+36*_C1^2*x^4-12*_C1^3*x^2+81)^(1/2))*x^2)^(1/3)+6*(1/3*x^2
-1/3*_C1)*x/((-108+12*(12*x^8-36*_C1*x^6+36*_C1^2*x^4-12*_C1^3*x^2+81)^(1/2))*x^
2)^(1/3))]